Problem: Rewrite as a simplified fraction. ${0.51\overline{7} = {?}}$
Solution: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 517.7777...\\ 100x &= 51.7777...\end{align*}}$ Subtracting the two equations, we have: ${900x = 466}$ ${x = \dfrac{466}{900}} $ ${x = \dfrac{233}{450}} $